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Dynamics Of Rigid Bodies2D

Three-Dimensional Kinetics of Rigid Bodies - Theory & Concepts - Gyroscopic

Study of the relationships between the forces and moments acting on rigid bodies and the resulting 3D motion, including Eulerian angles, gyroscopic motion, and torque-free motion.

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Gyroscopic Precession & Torque Dynamics

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Equations

Rotor Moment of Inertia:
Is=12mR2=0.5×1.5×0.42=0.120 kg⋅m2I_s = \frac{1}{2} m R^2 = 0.5 \times 1.5 \times 0.4^2 = 0.120\,\text{kg}\cdot\text{m}^2
Gravity Torque Magnitude:
τ=mgdsin⁡θ=1.5×9.81×0.35×sin⁡(60∘)=4.46 N⋅m\tau = m g d \sin\theta = 1.5 \times 9.81 \times 0.35 \times \sin(60^\circ) = 4.46\,\text{N}\cdot\text{m}
Spin Angular Momentum:
Hs=Isωs=0.120×20.0=2.40 kg⋅m2/sH_s = I_s \omega_s = 0.120 \times 20.0 = 2.40\,\text{kg}\cdot\text{m}^2/\text{s}
Precession Rate (steady):
Ω=τHssin⁡θ=mgdIsωs=5.1502.40=2.15 rad/s\Omega = \frac{\tau}{H_s \sin\theta} = \frac{m g d}{I_s \omega_s} = \frac{5.150}{2.40} = 2.15\,\text{rad/s}
Precession Vector (Ω)
Angular Momentum (H)
Gravity Torque Vector (τ)

Concept: The gravitational force creates a torque τ\tau perpendicular to the spin axis.

Instead of falling down, the gyroscope precesses horizontally because the torque changes the direction of the spin angular momentum vector H\mathbf{H}.

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